Levinson's theorem for the nonlocal interaction in one dimension

Shi Hai Dong

doi:10.1023/A:1003636110510

Levinson's theorem for the nonlocal interaction in one dimension

Shi Hai Dong

Research output: Contribution to journal › Article › peer-review

11 Scopus citations

Abstract

The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm-Liouville theorem. The critical case where the Schrödinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n₊ (n_-) of bound states with even (odd) parity is related to the phase shift η₊(0)[η (0)] of the scattering states with the same parity at zero momentum as (equation presented) and (equation presented) The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.

Original language	English
Pages (from-to)	1529-1541
Number of pages	13
Journal	International Journal of Theoretical Physics
Volume	39
Issue number	6
DOIs	https://doi.org/10.1023/A:1003636110510
State	Published - Jun 2000
Externally published	Yes

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@article{52f19705f9584b48837e45318de61554,

title = "Levinson's theorem for the nonlocal interaction in one dimension",

abstract = "The Levinson theorem for the one-dimensional Schr{\"o}dinger equation with both local and the nonlocal symmetric potentials is established by the Sturm-Liouville theorem. The critical case where the Schr{\"o}dinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n+ (n-) of bound states with even (odd) parity is related to the phase shift η+(0)[η (0)] of the scattering states with the same parity at zero momentum as (equation presented) and (equation presented) The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.",

author = "Dong, {Shi Hai}",

note = "Funding Information: S.H.D. is grateful to Prof. Mark S. Child for an invitation to visit the University of Oxford. This work was supported in part by the Royal Society and in part by Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, US Department of Energy.",

year = "2000",

month = jun,

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journal = "International Journal of Theoretical Physics",

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T1 - Levinson's theorem for the nonlocal interaction in one dimension

AU - Dong, Shi Hai

N1 - Funding Information: S.H.D. is grateful to Prof. Mark S. Child for an invitation to visit the University of Oxford. This work was supported in part by the Royal Society and in part by Division of Chemical Sciences, Office of Basic Energy Sciences, Office of Energy Research, US Department of Energy.

PY - 2000/6

Y1 - 2000/6

N2 - The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm-Liouville theorem. The critical case where the Schrödinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n+ (n-) of bound states with even (odd) parity is related to the phase shift η+(0)[η (0)] of the scattering states with the same parity at zero momentum as (equation presented) and (equation presented) The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.

AB - The Levinson theorem for the one-dimensional Schrödinger equation with both local and the nonlocal symmetric potentials is established by the Sturm-Liouville theorem. The critical case where the Schrödinger equation has a finite zero-energy solution is also analyzed. It is shown that the number n+ (n-) of bound states with even (odd) parity is related to the phase shift η+(0)[η (0)] of the scattering states with the same parity at zero momentum as (equation presented) and (equation presented) The problems on the positive-energy bound states and the physically redundant state related to the nonlocal interaction are also discussed.

UR - http://www.scopus.com/inward/record.url?scp=0034350051&partnerID=8YFLogxK

U2 - 10.1023/A:1003636110510

DO - 10.1023/A:1003636110510

M3 - Artículo

SN - 0020-7748

VL - 39

SP - 1529

EP - 1541

JO - International Journal of Theoretical Physics

JF - International Journal of Theoretical Physics

IS - 6

ER -

Levinson's theorem for the nonlocal interaction in one dimension

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